Apparent Universal Behavior in 2<sup>nd</sup> Moments of Random Quantum Circuits
ORAL
Abstract
Just how fast does the brickwork circuit form an approximate 2-design? Is there any difference between anticoncentration and being a 2-design? Does geometry matter? How deep a circuit will I need in practice? We tell you everything you ever wanted to know about second moments of random quantum circuits, but were too afraid to ask. Our answers are generally based on numerical results on up to 50 qubits.
Our first contribution is a strategy to determine explicitly the optimal experiment which distinguishes any given ensemble from the Haar measure. With this formula and some computational tricks, we are able to compute $t = 2$ multiplicative errors exactly out to modest system sizes. As expected, we see that most families of circuits form $\epsilon$-approximate $2$-designs in depth proportional to $\log n$. For the 1D brickwork, we work out the leading-order constants explicitly. For graph-sampled architectures, we find some exceptions which are much slower, proving that they require at least $\Omega(n)$ gates per site. This answers a question asked by ref. [Mittal2023] in the negative. We explain these exceptional architectures in terms of connectedness, corresponding loosely to a separation of timescales. Based on this intuition we conjecture universal upper and lower bounds for graph-sampled circuit ensembles.
For many architectures, the optimal experiment which determines the multiplicative error corresponds exactly to the collision probability (i.e. anticoncentration). However, we find that the star graph anticoncentrates much faster than it forms an $\epsilon$-approximate $2$-design. Finally, we show that one needs only ten to twenty layers to construct an approximate $2$-design for realistic parameter ranges. This is a large constant-factor improvement over previous constructions. We find that the parallel complete-graph architecture is not quite optimal, partially resolving a question raised by ref. [Dalzell2022].
Our first contribution is a strategy to determine explicitly the optimal experiment which distinguishes any given ensemble from the Haar measure. With this formula and some computational tricks, we are able to compute $t = 2$ multiplicative errors exactly out to modest system sizes. As expected, we see that most families of circuits form $\epsilon$-approximate $2$-designs in depth proportional to $\log n$. For the 1D brickwork, we work out the leading-order constants explicitly. For graph-sampled architectures, we find some exceptions which are much slower, proving that they require at least $\Omega(n)$ gates per site. This answers a question asked by ref. [Mittal2023] in the negative. We explain these exceptional architectures in terms of connectedness, corresponding loosely to a separation of timescales. Based on this intuition we conjecture universal upper and lower bounds for graph-sampled circuit ensembles.
For many architectures, the optimal experiment which determines the multiplicative error corresponds exactly to the collision probability (i.e. anticoncentration). However, we find that the star graph anticoncentrates much faster than it forms an $\epsilon$-approximate $2$-design. Finally, we show that one needs only ten to twenty layers to construct an approximate $2$-design for realistic parameter ranges. This is a large constant-factor improvement over previous constructions. We find that the parallel complete-graph architecture is not quite optimal, partially resolving a question raised by ref. [Dalzell2022].
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Presenters
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Daniel Belkin
- University of Illinois Urbana-Champaign